Representation of Vectors

$B\,$ is called the terminal point or the head of the vector.
memory device: a creature should lead with its head, not its tail

For ease of language, most people blur the distinction between a vector and the arrow representing it.
Thus, we can say the ‘tail of a vector’ instead of the (more correct) ‘tail of the arrow representing the vector’.

The vector from an initial point $\,A\,$
to a terminal point $\,B\,$ can be notated by $\,\overrightarrow {AB}\,$.
$\overrightarrow{AB}\,$ can be read aloud as ‘vector $\,A\,$ $\,B\,$’.

A single letter can be used to represent a
vector.
There are several conventional options for notation:

put an arrow over the top: $\,\vec v\,$

make the letter bold in a non-italic typeface: $\,\boldsymbol{\rm v}\,$

make the letter bold in an italic typeface: $\,\boldsymbol{v}\,$

When hand-writing, use $\,\vec v\,$, since it's difficult to distinguish bold from non-bold in handwriting.
Note that the arrow used in the notation for vectors is a reminder that direction is important!

The letter ‘v’ is commonly used to represent a vector ($\,\vec v\,$, $\boldsymbol{\rm v}\,$, $\,\boldsymbol{v}\,$),
since it's the
first letter in the world vector.

When hand-writing, you often put only a half-arrow over the top, like this: $\,\overset{\rightharpoonup}{\smash{v}\vphantom{b}}\,$
It's quicker and easier than making the full arrowhead.

When vectors are used together with scalars, it's important to tell them apart.
Compare the notations (say) $\,k\,\vec v\,$ with $\,k\,\boldsymbol{\rm v}\,$.
Dr. Burns prefers the notation $\,k\,\vec v\,$ as easiest to tell apart.

Size (Magnitude) of a Vector

The size (magnitude) of a vector
is the length of the arrow representing it.

The vectors shown above are all different,
because they all have different directions.
However, they all have the same size, which is:
$\sqrt{2^2 + 5^2} = \sqrt{29}$

The size of a vector is the length of the arrow representing it.

In common usage, the words size, magnitude, and length are all used interchangeably in this context.
So, these are all the same:

the size of a vector

the magnitude of a vector

the length of a vector

The size of a vector is notated by putting the vector inside vertical bars.
For example:
The size of vector $\,\overrightarrow{AB}\,$ is notated by $\,\|\overrightarrow{AB}\|\,$.
The size of vector $\,\vec v\,$ is notated by $\,\|\vec v\|\,$.

Note:

The size of a real number $\,x\,$ is denoted using absolute value: $\,|x|\,$

The size of a vector $\,\vec v\,$ is denoted by $\,\|\vec v\|\,$.

Some people use ‘$\,|\cdot|\,$’ for both real numbers and vectors.
This is called ‘operator overloading’ in computer science.
For beginning math students, it's easiest (and safer)
to use different notation.

Since size gives the length of an arrow,
it is a nonnegative quantity:

For all vectors $\,\vec v\,$, $\,\|\vec v\| \ge 0\,$.

The zero vector is the unique vector of length zero.
If a vector has length zero, then it is the zero vector.
That is: